Let \(\mathbb {Q}\)  be the reduced bar resolution \(\mathcal {B}\)  and let \(\mathbb {P}\)  be the resolution \(\mathbb {K}\) . We can take \(\Phi _{K}^{B} := \iota ,\)  the map defined just before Equation (REF ). It is not obvious how to define \(\Phi _{B}^{K}\) , the chain map from the reduced bar resolution to the resolution \(\mathbb {K}\) . The method of homotopy lifting overcomes this barrier because it does not require definining the chain \(\Phi _{B}^{K}\) . We now recall important notions about the homotopy lifting technique given in [1]} and [2]}.
